The timelike incompleteness criterion for CMC slices

Let $\mathcal{V}$ be a cosmological spacetime. A timelike path is incomplete when it has finite length. Timelike incompleteness criterion. If every future or every past timelike path in $\mathcal{V}$ is incomplete, then there exists a CMC slice. This is presented as the difficult part of the ray-and-line criterion and is related to the idea that a global singularity should force a CMC slice. The claim is known for certain crushing and strong-curvature singularities, but remains open for general cosmological spacetimes.